Generalized Quadrangles and Transitive Pseudo-Hyperovals

A pseudo-hyperoval of a projective space PG(3n− 1, q), q even, is a set of q +2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudo-hyperoval with an irreducible transitive stabiliser is elementary. We then deduce from this result a classification of the thick generalised quadrangles Q that admit a point-primitive, line-transitive automorphism group with a pointregular abelian normal subgroup. Specifically, we show thatQ is flag-transitive and isomorphic to T ∗ 2 (H), whereH is either the regular hyperoval of PG(2, 4) or the Lunelli–Sce hyperoval of PG(2, 16).